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Bulk-boundary decomposition of neural networks

arXiv 2511.02003 ↗First posted 3 Nov 2025cs.LG cond-mat.dis-nn hep-phcited by 0

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Abstract (arXiv)

We present the bulk--boundary decomposition as a new framework for understanding the training dynamics of deep neural networks. Starting from the stochastic gradient descent formulation, we show that the Lagrangian can be reorganized into a data-independent bulk term and a data-dependent boundary term. The bulk captures the intrinsic dynamics set by network architecture and activation functions, while the boundary reflects stochastic interactions from training samples at the input and output layers. This decomposition exposes the local and homogeneous structure underlying deep networks. As a physical consequence of locality and homogeneity, we derive the energy continuity equation within a deep neural network.

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References · 26

  1. 2017Deep Neural Networks as Gaussian Processescited by 617
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14 more
  1. 2021Nonperturbative renormalization for the neural network-QFT correspondencecited by 24
  2. 2023Neural Network Field Theories: Non-Gaussianity, Actions, and Localitycited by 21
  3. 2021Unified field theoretical approach to deep and recurrent neuronal networkscited by 20
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  7. 2022A duality connecting neural network and cosmological dynamicscited by 10
  8. 2024Conformal Fields from Neural Networkscited by 10
  9. 2021The Hintons in your Neural Network: a Quantum Field Theory View of Deep Learningcited by 9
  10. 2024Bayesian RG Flow in Neural Network Field Theoriescited by 9
  11. 2025Applications of Statistical Field Theory in Deep Learningcited by 8
  12. 2024Emergent field theories from neural networkscited by 4
  13. 2025Revisiting Deep Information Propagation: Fractal Frontier and Finite-size Effectscited by 2
  14. 2024Dynamic neuron approach to deep neural networks: Decoupling neurons for renormalization group analysiscited by 1

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